Harmonic functions

Abstract

This part begins with a review of basic properties of harmonic functions, whose consideration is suggested by several branches of physics (Newtonian gravity, electrostatics) and mathematics (e.g. complex analysis and the theory of surfaces). When the theory of surfaces was developed, the Laplace-Beltrami operator was viewed as a differential parameter of order 2, and this is shown in detail. The reader is then introduced to the theory of distributions and Sobolev spaces, while regularity theory is introduced through the Caccioppoli derivation of integral bounds on solutions of linear elliptic equations. The concept of ellipticity is then defined in various cases of interest. After an outline of spectral theory, the De Giorgi example of Laplace equation with mixed boundary conditions is studied. As a next step, Morrey and Campanato spaces, and functions of bounded mean oscillation, are studied. Part II ends by focusing on pseudo-analytic functions, with the associated generalized form of Cauchy-Riemann systems, and some remarkable properties of biharmonic and polyharmonic functions.